Although computer-assisted, corneal topography analysis systems are useful clinical tools that have advanced the understanding of normal and abnormal corneal shapes, the investigation of corneal topography suffers from undefined relationships between videokeratoscope power and elevation and the elevation of the real corneal surface and the lack of a compact, numerical representation of topography. These inadequacies limit the physical and optical relevance and prevent the statistical analysis of videokeratoscope output. A valid method for estimating and compactly representing the corneal surface is necessary as a scientific foundation for this growing research field. We developed the Corneal Visualization Tool (CVT), an interactive computer graphics program for measuring, modifying, and generating polar coordinate arrays of power and viewing color-coded isodiopter maps. The CVT prototype reduces empirical power arrays to measurements of the patter of the distribution of power along semimeridians and the azimuthal extrema specified by the user. We used the CVT prototype to explore relationships between power arrays and maps. We propose to apply geometric principles to the output of a videokeratoscope and the physical positions of videokeratoscope optical components to estimate the corneal profile and the possible range of the profile along each semimeridians. The profiles, arranged in polar arrays of elevations, will be our estimate of the surface. The procedure will be empirically evaluated using test surfaces with specified aspheric profiles. We will further develop CVT to fit a series of mathematical expressions to the corneal profile, respecting the possible range of the profile, and represent a cornea's surface by term coefficients or their extrema. We will develop a method for measuring and displaying the differences between elevation arrays. We will map the azimuthal functions of CVT representations into feature space to develop and alternative format the statistical pattern analysis. We will collect clinical data bases to assess the applicability of our procedures to diverse clinical conditions, the effect of videokeratoscope repeatability ion our procedures, and CVT's utility for discriminating between clinical classes. Our proposal directly addresses the identification of a theoretically sound mathematical foundation for the scientific evaluation of corneal topography.